The paintings of El Greco, 'The Greek', may well be the strangest in all art. Even close familiarity can't change that. They never lose their power to amaze and the more you look, the more the late visions stream from your grasp - into that crackling, flickering world of acid colours exploding out of seething darkness, of warped and buckling space, of violently elongated figures, of saints shooting to heaven like embers up a chimney, of ectoplasmic clothes and bodies like molten beeswax. Why did he paint this way? Where did these images come from? And into what sort of mind? El Greco's art is so incorrigibly weird as always to be raising the question of its own strangeness.

He was born Domenikos Theotocopoulos in Crete in 1541 and if he'd never left we would know nothing about him. For until the age of 26, when he set sail for Italy, and eventually Spain, El Greco painted only icons.

There is one in this show, from a church on the hot and windy island of Syros, on which his signature was recently uncovered. A tiny gilded panel, glowing in the dark, it shows the last moments of the Virgin. Instantly you discover the future El Greco in its crumpled space, in the updraft of angels rushing to God, in the cut-out figures and the collage effects, in the ultra-bright highlights that zigzag unnaturally up the painting. It is the very first exhibit and perhaps you will carry it in your mind to the last - from its golden light to the final electric sparks - but El Greco's style is not merely adapted from icons.

In Venice, he looked long and hard at Titian and Tintoretto, that much is clear, learning about volume, colour, movement and light, about staging dramatic perspectives. Two years later, in Rome, he immediately recognises the magnificence of Michelangelo's figures, in their rhythmic ecstasies and throes, even though he is supposed to have caused a cata clysmic furore (followed by his swift removal to Spain) by casually observing that Il Gigante couldn't paint.

The sheer jump-cut speed of El Greco's transformations is made brilliantly clear in this show. He will paint an exceptionally moving Pietà, based on Michelangelo's sculpture in Rome. Or a Boschian Leviathan ingesting devils. Or an image of Christ (once owned by Delacroix) about to be stripped of his blood-red robes by a phalanx in cold sharp metal, the air clanging with threat, superb as anything by Titian. And all this in the same few seasons.

El Greco was looking for work and being rebuffed over and again, sacked and even ejected from his lodgings in Rome (no one knows why), failing to impress Philip II of Spain who commissioned but then rejected a work, settling in Toledo (nobody knows the reason).

But this defiant outsider who signed his name in Greek to the last wasn't just adjusting to the different tastes of his patrons. To witness a really extraordinary crystallisation - El Greco becoming El Greco, as it were - look at all the different versions assembled here of The Purification of the Temple . It's the same basic scene: Christ taking his flail to the money-lenders in a neo-classical building. Fairly naturalistic to begin with, excepting one of El Greco's supercharged skies, it soon grows eerily outlandish. A group of pop-up heads appears bottom right - including a replica Titian self-portrait inserted like a scrap into an album. Christ's eyes become as primitive and alien as a Phoenician eagle. His sweeping motion corkscrews, then torques, then swirls upwards as his body lifts off. All around him sinners shiver and ripple, shielding their eyes from the dazzle of his inner brilliance.

It is as if an icy wind was blowing through these pictures. Space flattens, daylight disappears, figures, whipped into abnormal extrusions, no longer cast any shadows. In the last one, after 1600, even a statue in its niche has become authentic El Greco, miles of boneless body between pinhead and feet, no longer static but gyring in the darkness.

How to account for it? The modernists didn't feel the need to bother, since it looked so obviously like proto-modern art. You can see what they took from El Greco - Cézanne's late and wavering bathers, Picasso's anatomical distortions, the no-space, no-perspective chopped viewpoints of cubism. You could, and people sometimes do, find elements of surrealism in El Greco too. But it's a skewed view that would estimate him in the supporting role of precursor to the moderns.

Others are content with El Greco as the mannerist who left Italy when the style was quite new and believed for several decades, like the marooned World War II soldier, that it was all still raging. Certainly he saw and admired Parmigianino's swan-necked saints and certainly the worst of El Greco - the Bambi-eyed apostles, the Pinocchio angels - looks like a reductio ad absurdum.

But mannerism has not anything more extreme to show than El Greco's Immaculate Conception, for one, in which the Virgin, elongated to extortionate proportions, is raised to a sun-and-moonlit ecstasy by an angel, one of whose gigantic wings, seen side on, comes at you like a slicing machine.

An art that demands explanation - that's been the line since El Greco was rediscovered in the nineteenth century. He was mad, or maddened by hash, or suffering severe hallucinations. He was a supreme visionary or his vision was distorted by degenerative astigmatism. That last, and most ridiculous, is supposed to account for his very queer optics and forms. But El Greco can go in and out focus, in and out of realism, all in one painting.

Besides which there are the portraits, so piercingly expressive and clear - the inquisitor in his spectacles, tightly gripping his chair, the elderly ascetic, so alive with quizzical wisdom. It is no surprise that Velázquez revered them and their genius is evident even in El Greco's imaginary portraits of saints: Francis contemplating a crucifix as tenderly as a mother her newborn, Peter patiently remonstrating with a gaunt and emphatic Paul.

But there is no such tenderness in El Greco's last grand visions. To come across these paintings in the midnight shadows of the Sainsbury Wing - no show has ever been better presented - is to see vast screens of action flaring in the darkness. The Opening of the Fifth Seal rippling like St Elmo's fire (who knows what's going on here?). The neck-breaking Resurrection in which Christ soars way above you and the soldiers, amazed, seem to be catching light.

It is often said that El Greco's art is all about dematerialisation, about the body becoming the spirit. Perhaps that is true and goes to his religious beliefs, although we know scarcely anything about them. But these late works are as other-worldly as El Greco's first icons, as if he had diverted their transcendental power to his own singular art. An art in which exultation is very close to fear, and which envisages the divine as irreducibly supernatural. As supernatural as the infant Christ in a nativity El Greco painted for his own tomb, blazing with mineral aura. From which the shepherds draw back, astonished, as if from a ticking bomb: a revelation both shocking and awesome.

Three to see

Philip Guston Royal Academy, London W1, until 12 April Tremendous retrospective of the great tragi-comedian of American painting

Vuillard Royal Academy, London W1, until 12 April Infinite riches in little rooms: French intimisme at its best